Download Finding Square Roots 7.1

7.1
Finding Square Roots
How can you find the dimensions of a square
or a circle when you are given its area?
When you multiply a number by itself, you square the number.
Symbol for squaring
is the exponent 2.
⋅
42 = 4 4
= 16
4 squared is 16.
To “undo” this, take the square root of the number.
—
—
Symbol for square root
—
is a radical sign, √ .
1
√ 16 = √ 42 = 4
The square root of 16 is 4.
ACTIVITY: Finding Square Roots
Work with a partner. Use a square root symbol to write the side length of
the square. Then find the square root. Check your answer by multiplying.
—
a. Sample: s = √ 121 = 11 ft
Area ä 121 ft 2
Check
11
× 11
s
11
110
The side length of the
square is 11 feet.
b.
Area ä 81 yd2
121
s
c.
Area ä 324 cm 2
d.
s
s
✓
Area ä 361 mi 2
s
COMMON
CORE
Square Roots
In this lesson, you will
● find square roots of
perfect squares.
● evaluate expressions
involving square roots.
● use square roots to
solve equations.
Learning Standard
8.EE.2
s
s
e.
Area ä 225 mi 2
s
f.
Area ä 2.89 in. 2
s
s
288
Chapter 7
Real Numbers and the Pythagorean Theorem
s
g.
Area ä
s
s
s
4 2
ft
9
2
ACTIVITY: Using Square Roots
Work with a partner. Find the radius of each circle.
b.
a.
r
r
Area ä Ĭ yd 2
Area ä 36 Ĭ in.2
c.
d.
r
r
Area ä 0.25 Ĭ ft 2
3
Math
Practice
Calculate
Accurately
How can you use
the graph to help
you determine
whether you
calculated the
values of T
correctly?
Area ä
9
Ĭ m2
16
ACTIVITY: The Period of a Pendulum
Work with a partner.
The period of a pendulum is the time (in seconds)
it takes the pendulum to swing back and forth.
—
L
The period T is represented by T = 1.1√ L , where
L is the length of the pendulum (in feet).
Copy and complete the table. Then graph the
function. Is the function linear?
L
1.00
1.96
3.24
4.00
4.84
6.25
7.29
7.84
9.00
T
4. IN YOUR OWN WORDS How can you find the dimensions of a square
or a circle when you are given its area? Give an example of each.
How can you check your answers?
Use what you learned about finding square roots to complete
Exercises 4–6 on page 292.
Section 7.1
Finding Square Roots
289
7.1
Lesson
Lesson Tutorials
Key Vocabulary
A square root of a number is a number that, when multiplied by itself,
equals the given number. Every positive number has a positive and a
negative square root. A perfect square is a number with integers as its
square roots.
square root, p. 290
perfect square, p. 290
radical sign, p. 290
radicand, p. 290
EXAMPLE
Finding Square Roots of a Perfect Square
1
Find the two square roots of 49.
⋅
Study Tip
⋅
7 7 = 49 and (−7) (−7) = 49
Zero has one square
root, which is 0.
So, the square roots of 49 are 7 and −7.
—
The symbol √ is called a radical sign. It is used to represent a square
root. The number under the radical sign is called the radicand.
—
Positive Square Root, √
Negative Square Root, −√
—
2
—
Both Square Roots, ±√
—
√ 16 = 4
EXAMPLE
—
—
−√ 16 = −4
±√ 16 = ±4
Finding Square Roots
Find the square root(s).
—
—
√ 25 represents the
a. √ 25
positive square root.
—
—
Because 5 = 25, √ 25 = √ 5 = 5.
2
√
2
√
—
9
− — represents the
—
9
16
b. − —
—
√
()
√( )
3 2
9
9
= —, − — = −
4
16
16
Because —
—
3 2
3
= − —.
4
4
16
negative square root.
—
—
±√ 2.25 represents both the positive
and the negative square roots.
—
c. ±√ 2.25
—
—
Because 1.52 = 2.25, ±√ 2.25 = ±√ 1.52 = 1.5 and −1.5.
Find the two square roots of the number.
Exercises 7–18
1. 36
2.
100
3.
121
5.
√
6.
√ 12.25
Find the square root(s).
—
4. −√ 1
290
Chapter 7
—
4
25
± —
Real Numbers and the Pythagorean Theorem
—
Squaring a positive number and finding a square root are inverse operations.
You can use this relationship to evaluate expressions and solve equations
involving squares.
EXAMPLE
Evaluating Expressions Involving Square Roots
3
Evaluate each expression.
—
a. 5√ 36 + 7 = 5(6) + 7
1
4
—
√
18
2
Evaluate the square root.
= 30 + 7
Multiply.
= 37
Add.
—
1
4
b. — + — = — + √ 9
Simplify.
1
4
=—+3
Evaluate the square root.
1
4
= 3—
c.
Add.
2
( √—
81 ) − 5 = 81 − 5
Evaluate the power using inverse operations.
= 76
EXAMPLE
4
Subtract.
Real-Life Application
The area of a crop circle is 45,216 square feet. What is the radius
of the crop circle? Use 3.14 for 𝛑 .
A = πr2
Write the formula for the area of a circle.
45,216 ≈ 3.14
3.14r 2
Substitute 45,216 for A and 3.14 for π.
14,400 = r 2
Divide each side by 3.14.
—
—
√ 14,400 = √ r 2
Take positive square root of each side.
120 = r
Simplify.
The radius of the crop circle is about 120 feet.
Evaluate the expression.
Exercises 20–27
—
7. 12 − 3√ 25
√
—
8.
28
7
— + 2.4
9.
—
15 − ( √ 4 )
2
10. The area of a circle is 2826 square feet. Write and solve an
equation to find the radius of the circle. Use 3.14 for π.
Section 7.1
Finding Square Roots
291
Exercises
7.1
Help with Homework
1. VOCABULARY Is 26 a perfect square? Explain.
2. REASONING Can the square of an integer be a negative number? Explain.
—
3. NUMBER SENSE Does √ 256 represent the positive square root of 256, the
negative square root of 256, or both? Explain.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Find the dimensions of the square or circle. Check your answer.
4.
5.
Area ä 441 cm 2
6. Area ä 64 Ĭ in.2
Area ä 1.69 km 2
r
s
s
s
s
Find the two square roots of the number.
1
7. 9
8. 64
9. 4
10. 144
Find the square root(s).
—
√
12. ±√ 196
—
—
1
961
13. ± —
—
15. ±√ 4.84
√
—
—
2 11. √ 625
—
16. √ 7.29
—
17. −√ 361
19. ERROR ANALYSIS Describe and correct
the error in finding the square roots.
✗
9
100
14. − —
18. −√ 2.25
±
√
—
1
4
1
2
—=—
Evaluate the expression.
3 20.
—
2
(√9 )
—
+5
24. √ 6.76 + 5.4
—
2
21. 28 − ( √ 144 )
—
25. 8√ 8.41 + 1.8
√
—
—
1
16
22. 3√ 16 − 5
23. 10 − 4 —
(√
(√
—
26. 2
80
— −5
5
)
28. NOTEPAD The area of the base of a square notepad is
2.25 square inches. What is the length of one side of
the base of the notepad?
—
27. 4
147
3
— +3
r
29. CRITICAL THINKING There are two square roots of 25.
Why is there only one answer for the radius of the button?
2
A â 25mm
Ĭ
292
Chapter 7
Real Numbers and the Pythagorean Theorem
)
Copy and complete the statement with <, >, or =.
—
30. √ 81
31. 0.5
8
—
—
√
3
32. —
2
√ 0.25
25
4
—
1
2
33. SAILBOAT The area of a sail is 40 — square feet. The base
and the height of the sail are equal. What is the height of
the sail (in feet)?
34. REASONING Is the product of two perfect squares always
a perfect square? Explain your reasoning.
h
35. ENERGY The kinetic energy K (in joules) of a falling apple
v2
2
is represented by K = —, where v is the speed of the
b
apple (in meters per second). How fast is the apple
traveling when the kinetic energy is 32 joules?
Area â 4 Ĭcm2
36. PRECISION The areas of the two
watch faces have a ratio of 16 : 25.
a. What is the ratio of the radius of
the smaller watch face to the radius
of the larger watch face?
b. What is the radius of the larger
watch face?
37. WINDOW The cost C (in dollars) of making a square window with a side length
n2
5
of n inches is represented by C = — + 175. A window costs $355. What is
the length (in feet) of the window?
38.
The area of the triangle is represented by the
———
formula A = √ s(s − 21)(s − 17)(s − 10) , where s is equal
17 cm
10 cm
to half the perimeter. What is the height of the triangle?
21 cm
Write in slope-intercept form an equation of the line that passes through the
given points. (Section 4.7)
39. (2, 4), (5, 13)
40. (−1, 7), (3, −1)
41. (−5, −2), (5, 4)
42. MULTIPLE CHOICE What is the value of x? (Section 3.2)
A 41
○
B 44
○
C 88
○
D 134
○
84í
(x à 8)í
xí
Section 7.1
Finding Square Roots
293